why don't the angle-side-side congruence theroem exist?

Answer 1

because that doesn't prove congruence
it's possible that 2 different triangles can produced with a given angle and then 2 sides. to show this, first draw an isosceles triangle. then extend the base past one of the bottom angles a bit. now connect the vertex of the orginal triangle to the end of the line that you just drew off the base. so now you see 2 triangles within one bigger one. the big one and one of the smaller ones share an angle and a side. so those are congruent. and since the original was isosceles, they also have another pair of congruent sides. but clearly they are not congruent triangles so the SSA rule doesnt exist

Answer 2

Because teachers don't want you using the abbreviation for it.

Actually because there can be 2 different triangles with those the same. I wish I could draw it because it would be easy to see...

Picture a right triangle ABC, with right angle at A. Now draw the line from B to the middle of AC and call this point D.

Extend line AC out past A. Now take a compass, put the needle at B and the pencil at D and draw a circle. This will hit the extended line AC at a point E. Triangles BED and BCD both share angle E and side BC, and BE = BD, so they have , uh, ***, but are not congruent.

Answer 3

It does. Under certain conditions.

If you have angle-side-SIDE, where the second side is longer than the first, there is an angle-side-side congruence theorem. But you'd generally list it as SsA so your teacher doesn't get mad that you spelled ***.